Symmetries and Group Theory - Lecture 2

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1 Symmetries and Group Theory - Lecture 2 1 Introduction Physics attempts to look for the global aspects of a system, since much of system behavior can be understood from general principles without investigating the details. It is well known that conservation of energy and momentum completely describe the kinematic motion of a system. Other conservation principles and symmetries are also inserted in the mathematical description of interactions and physical laws. There are many types of symmetries, some of these are geometrical and some more mathmatical, but all symmetries lead to conservation principles - the Noether theorem. There are discrete and continuous symmetries, but they must be coded into a mathematical form, or applied in a way to represent a symmetry. Thus the symmetry of baryon conservation requires that any reaction involving baryons must balance before and after a reaction. As an example, an equation for pion production in proton-proton collisions is written; P + P = P + P + π The number of baryons to the left and right of the equal sign is the same, as baryon number is conserved. This demonstrates that a pion, π, is not a baryon but it is a hadron (strongly interacting particle like the ptoton). Obviously, hadron number is not conserved. However, the reaction must also conserve other quantities such as energy, momentum, and charge. Electric charge is a mathematical symmetry and energy and momentum are related to time and sparial translations (geometric symmetries). There are 2 units of positive charge on the left of the equation and there must be 2 units on the right. Thus the pion in this reaction must be charge neutral. The mathematical tool for investigating symmetry is group theory. 2 The Noether theorem Although one does not need to invoke the Noether theorem to consider simple, particularly classical symmetries, it is a powerful tool in quantum theory, and for the physical interpretation of a symmetry. The theorem is stated in terms of the Lagrangian formulation of an interaction. Noether Theorem For every continuous variation of the action which leaves it invarient, there is a conserved quantity (a currrent) 1

2 An example of a conserved quantity is the charge of a system. This is mathematically stated by the equation of continuity, ρ t = ( V ρ) In this equation, ρ is the charge density and V the velocity of an incremental volume of charge. The 3-vector of the charge current is J = ρ V. Now consider the relativistic 4-vector in space-time. j = ( J, ρ) Operate on this vector by the 4-gradient, (, t ), to obtain; J + ρ t = 0 This states that the 4-current does not change in space-time so it is conserved, representing of course, conservation of charge. To demonstrate the Noether theorem, write a Lagrangian density as, L(φ k (x), uφ k (x) ). In x u this expression, φ k, uφ u are the fields, which for example, give the position and velocity x u amplitudes. The Lagrangian of this system is; L = d 3 x L t 2 The local action is S = dt L. The equations of motion follow from stationary action. t 1 δ S = 0 This introduces the Euler-Lagrange equations by applying the calculus of variations. Then look at changes to these equations due to small changes in the fields. This produces a conserved current. Without developing a general proof which takes some effort, look at the simple example of energy conservation. In this case there is a particle of mass, m moving under the influence of a potential, V. The action is; S = dt L[x(t), ẋ(t)] = dt[(m/2) ẋ 2 i V (x)] Let Q = t. For example Q[x] = ẋ. Then; QL = m ẋ i ẍ i V x i ẋ i = d dt [m/2 ẋ 2 V ] = d dt [F] 2

3 V(x) L Figure 1: A figure showing the potential function, V (x), of a 1-D crystal lattice which has a symmetry upon translation by the lattice dimensions, L x Here, F = [m/2 ẋ 2 V ]. Set j = L ẋ Q[x] f j = m ẋ 2 F j = m/2 ẋ 2 + V (x) This is an expression for the energy of the system. Thus if the energy is conserved (current) then the system is invariant under a time translation. 3 Examples of discrete symmetries As a simple example consider the potential well of a 1-D atomic crystal lattice. This has a symmetry about the position of the atoms in the lattice as shown on Figure 1. A translation by the lattice dimensions produces the same potential. This is an example of a discrete, geometric symmetry. In this example there is a translation operator T n (L) which operates on a state vector of a quantum system changing the position vector from x to x + nl, where n is a positive or negative integer. Therefore if ψ(x) is a state vector, then; ψ(x + nl) = T n ψ(x) The state vector ψ(x) and ψ(x + nl) are equivalent descriptions of the system. Therefore the expectation values of an operator A in this space is the same using either state vector. ψ A ψ = T n ψ A T n ψ This can be written; 3

4 Hypercharge π K K 0 K + π 0 π + K Charge Figure 2: A figure showing a discrete symmetry among different elementary particles in the 2-D space of charge vs hypercharge ψ A ψ = ψ (Tn 1 A T n) ψ Then A = (Tn 1 A T n ). Therefore the invarience of the potential operator is obvious; ψ V (x) ψ = ψ (Tn 1 V (x) T n) ψ = V (X + nl) The Hamiltonian H = p 2 /2m+V commutes with the operator T n since a spatial translation does not change the value of p. The above clearly connects the symmetry to the mathematical description of the state. In another example, this time from particle physics, we have already observed a symmetry in charge conservation and baryon number. However, a pion is not a baryon. We introduce another quantum number called hypercharge, Y, which is baryon number plus a strangeness quantum number, Y = B + S. Strangeness is to be associated with a strange quark which produces an additional hadronic meson called a kaon. The symmetry of the pion/kaon system is discrete in a 2-D plot of charge vs hypercharge, as shown in Figure 2. It is an example of a mathematical symmetry which underlies the strong interaction. This figure suggests that there is an underlying symmetry between mesons under the strong interaction. In fact the splitting along rows is due to the electromagnetic and not the strong interaction. The above symmetry is due to a gauge transformation and is developed later, but a gauge transformation leaves a Lagrangian invarient as did the simple case demonstrated above. 4

5 4 Groups We begin with the mathematical description of a group. This is introduced by a consideration of discrete, as opposed to continuous, group elements. To form a group, there is a set of elements and a algebra for combining these elements (e.g. multiplication ). The set forms a group if; 1. The algebra of combinng the elements of a group, a i, produces another element of the group. a i a j = a k. 2. There is an idenity element, I, such that, Ia i = a i 3. There is an inverse element, a 1, such that a i a i = I 4. All elements are mapped into all elements of the group by these operations 5. The algebra is associative. a i (b j c k ) = (a i b j )c k Any group may have different representations. For example the identity element is always mapped into itself. Then we classify group representations. An important class are homomorphic maps. These maps preserve the group algebra. Suppose the map takes the group F into the group G; F G with an element F(i) G(i), then apply the group multiplication. F(k k ) = F(f i )F(j k ) Applying the above map, if ; F(f k ) G(g i )G(g j ) = G(g k ) then the map is homomorphic. Mapping onto the identity is homomorphic but not always isomorphic. For the map to be isomorphic, the two groups must have a one-to-one correspondance between elements which also preserves the multiplication. f i g i f j g j f i f j = f k g i g j = g k An example of a simple group is C 2, represented by the multiplication Table 1 The numbers +1 and 1 form the elements of this group under normal multiplication. The order of a group is the number of group elements. Other representations of this group are the spatial inversion (parity) and a 2-element permutation. C 2 is the lowest order representation of the groups generated by cyclic permutations, C n. The cyclic group of order 3 has the multiplication table shown in Table 2. Geometrically, the rows and column represent angle 5

6 rotations of 2π/3, ie the verticies of an equilateral triangle. This group is referenced as the permutation or symmetric group. In classical physics, symmetry equations are applied to the solutions of a pde representing the physics. These solutions usually are eigenfunctions which can be represented as vectors in a linear, N-dimensional vector space. Group theory can then be realized as linear transformations on vectors. The multiplication of vectors and vector transformations is incorporated into matrix algebra. Thus we can use matrix representations for group elements. Then consider a group representation for the symmetric group in matrix form with element; ( ) cos(θi ) sin(θ i ) sin(θ i ) cos(θ i ) The group algebra is matrix multiplication. Thus the multiplication of two elements results in the sum of the two angles of the elements, θ i + θ j = θ k. ( ) ( ) cos(θi ) sin(θ i ) cos(θj ) sin(θ j ) = sin(θ i ) cos(θ i ) sin(θ j ) cos(θ j ) ( ) cos(θk ) sin(θ k ) sin(θ k ) cos(θ k ) The elements of the group, C 3 are then; ( ) ( ) ( ) 1 0 1/2 3/ /2 3/2 3/2 1/2 3/2 1/2 The first element is the identity, I. The next element represents a rotation by 2π/3, a 1 and the last element a rotation by 4π/3, a 2. Thus a 1 a 1 = a 2, a rotation by 4/pi/3 and a 1 a 2 = I, a rotation by 2π. Finally, a 2 a 2 = a 1, a rotation by 8π/3. Table 1: The multiplication table of the simple group of 2 elements, C 2 I a I I a a a I Table 2: The multiplication table of the simple group of 3 elements, C 3 I a b a b I b I a 6

7 Figure 3: The geometry showing the symmetries involved in obtaining the group elements of the Dihedral group D 3. There are 3 reflections about the diagonals and 2 rotations about the center of the figure. These symmetries plus the identity yield a total of 6 elements An abelian group has communicative multiplication, ab = ba. All cyclic groups are abelian. The smallest non-abelian group has order 6 and consist of reflections about the diagonals of an equilateral triangle and rotations about the center with angles of 2π/3 and 4π/3 3. This forms the Dihedral group, D 3 Table 3: The multiplication table of the Dihedral group of 6 elements, D 3 I a 1 a 2 a 3 a 4 a 5 a 1 I a 4 a 5 a 2 a 3 a 2 a 5 I a 4 a 3 a 1 a 3 a 4 a 5 I a 1 a 2 a 4 a 3 a 1 a 2 a 5 I a 5 a 2 a 3 a 1 I a 4 The lowest order of a non-cyclic group is 4. An example of the Dihedral group, D3 is shown by the group Table 3. The D 3 multiplication table shows 4 subgroups. A representation of one of these subgroups is a set of rotations by π/2, π, and 3π/2. In a set of matricies the representation elements are; ( ) ( ) ( ) ( ) 7

8 5 Group representations There are a number of ways a finite group may be represented. Supppose a group on a vector space V, and label a representation of that group by U. Let S be any non-singular operator (non-singular means the operator has an inverse) on V. Then another representation is; U = S 1 US U has the same dimension and the same algebra. We determine if two representations are equivalent by determining their character. If the representation is placed in matrix form, the group character is the trace of a matrix of an irreducible representation. χ = Tr [U] = i g ii The above is true because; Tr [S 1 US] = Tr [U] A representation is reducible if it can be represented in a basis as a matrix in the form, ( ) Ai (mm) B M i = i (mn) 0 C i (nn) The A, B, C, matricies have dimensions mxm, mxn, and nxn, respectively. For element multiplication; ( ) ( ) ( ) Ai (mm) B M i = i (mn) Aj (mm) B j (mn) Ai A = j A i B j + B i C j 0 C i nn 0 C j nn 0 C i C j Then operation on a vector; ( ) ( ) Ai B i U = 0 C i W ( Ai U + B i W C i W The W subspace is invarient if B 0. Thus the representation M = A C. Decomposition continues until irreducible representations are formed. Essentially one is finding the subspaces of othorgonal vectors which represent the group. There is then a set of such vectors for each irreducible representation of the group. There are nxn = n 2 possibilities for the matrix elements. Thus we expect; n 2 i = M ) In the above M is the dimension of the representation of the group. Consider the group D 3 (S 3 Figure 3. There are 3 classes in this group. 8

9 1. The identity matrix I 2. The elements a 1, a 2, a 3 3. The elements a 4 a 5 Suppose we have a matrix representation of the group as; ( ) D1 0 U 0 D 2 The group components D 1,2 are irreducible representations of U. Again suppose an operation on U by the operator S. One might suppose this represents a rotation in the vector space. Then the operation on the vectors (group elements) transforms them into other vectors in this space. If all the elements are reached by this transformation, then the group is irreducible, otherwise the group is reducible. In the above example D i represents subgroups if the identity is added to each subcomponent. In the above the D i are square matricies,and all essential properties of the group are contained in each representation. Suppose the following representation of the cyclic group, C 3. ( ) ( a ) ( a 2 ) A discete group has a finite number of classes. The group character of a class is unique to the class. The character of the group elements is obtained by forming the trace of the elements; χ(1) = 2, χ(2) = 1 + a, and χ(3) = 1 + a 2. The following is an equivalent representation. ( ) ( a ) ( a ) Now consider the permutation group, C 2, with multiplication Table 1 above. There are two group elements, the identity, I, and element a. From the multiplication table, there are two possible values for a, either ±1. These are the two inequivalent representations of the group C 2. The characters of the two representations are χ(1) = 2 and χ(2) = 0. The character is usually placed in tabular form - a character table, Table 4. The rows in the table are the ireducible representations and the columns are the classses The number of irreducible representations equals the number of classes. The permutation group has 3 classes, two being 1 dimensional and the third 2 dimensional. As a further example, consider the regular representation of the rotation group. A regular representation has the same dimensions as the order of the group. The rotation matricies are; 9

10 Identify the elements I with the first matrix, a with the second,and b with the third. This representation is reducible into 2 subgroups. The irreducible representations are simply the numbers; 1, e 2πi/3 e 4πi/3 or1, e 4πi/3, e 8πi/3 = e 2π/3 The second representation is interchange of elements 2 and 3. Multiplication by a complex number can be represented in matrix form. The first representation above can be written; ( ) ( ) 1 3 1/2 3 1 ( ) 1 3 1/2 3 1 This representation is irreducible. Note that the trace of all the matricies is the same. A crystallographic point group is composed of rotations, reflections, and inversions about a point. Every molecule has a point group association. The number of classes represents the subgroups or the number of symmetries of the system. We develop a group incorporating all these operations for an amonia molecule. The ammonia molecule has the S 3 (D 3 ) symmetry, as can be seen in Figure 4. This is the permutation group of the 3 hydrogen atoms. The representation of this group is irreducible and there are 3 group elements. The identity is a matrix with trace = 3. Then consider spatial reflections, for example in the (x, z) plane, This is obtained by the matrix; There are 2 other reflections each with trace = 1. There are also rotations with an element; Table 4: The character table of the permutation group C 3 after reducing the matrix to 1-D. The symbol w is the primitive third root of 1 (any complex number raised to the 3 power is 1) χ(1) χ(2) 1 a a 2 χ(3) 1 a 2 a 10

11 N z H H y x H Figure 4: The symmetry of the ammonia molecule representing 3 reflections and 3 rotations 1/2 3/2 0 3/2 1/ which has trace = 0. The character of the group is then (3, 1, 0). The number of classes are the number of elements connected by group multiplication to the element. To find group elements conjugate to an element, a, find the elements bab 1 for each element, b, in the group. All elements conjugate to each other form a class of the group. Consider the multiplication table for the group, D 3 (S 3 ), Table 3. The elements conjugate to a 1 are; a 2 a 3 a 4, The group character is the sum of the characters of the irreducible representations. Thus suppose there are g i elements in each class and g is the order of the group. Then the group character is g = χ i 2 The full group has the structure, M = I M R MI. For the ammonia example using the group S 4 ; S = [(2) 2 + (1) 2 ] = 5 There are 2 irreducible representations excluding the identity. Then to apply the symmetry to the description of the ammonia molecule, consider a real 3-D vector. This vector transforms using the 3-D matricies described above which is not the irreducible matricies of the group. However, a vector in 3-D must have a 3-D operator. Thus the matricies are composed of additive irreducible representations. For the vector there is the rotation group and the identity or M V = I M R with character [3, 0]. However, an axial vector has a reflection 11

12 transformation represented by the matrix; This has trace 1 be expressed by a 3-D. The group character contains the inversion and rotation groups, M AV = M I MR 6 Continuous groups In the previous sections we discussed groups with a discrete set of elements. Now cosnider groups with an infinite number of elements. Of these groups the Lie groups are of most interest. Suppose a transformation; x i = f i (x 1, x 2, x 3, θ) In the above, θ is a parameter. In a Lie group, the function, f, must be analytic in θ. Consider the set of NxN unitary matricies (a matrix when operating on a vector preserves its length). This set of matricies forms a group defined by U(n). They can ge considered rotation matricies. If we impose the condition that the determinant of the matricies is +1, we have have the special unitary group, SU(N). This is an NxN unitary matrix with N 2 1 independent parameters. For example, N = 2 means (2) 2 1 = 3 independent parameters. For N = 3 there are 8 parameters, sometimes called generators. In SU(2) the generators are, s k. In SU(3) the generators are M k. 7 Rotations Previously we looked at rotations and determined that finite rotations are not additive. Consider a rotation about the x axis as shown in Figure 5. Note that we are using a right handed system for coordinates and rotations. A rotation preserves the length of a vector (i.e It is a unitary operation.). Suppose we rotate a vector in 3-D space about the x axis with a rotation, ω. The coordinates (x, y, z) are rotated into (x, y, z ) by the matrix operation; x y x = A rotation about z is; cos(ω) sin(ω) 0 sin(ω) cos(ω) x y z = R x x y z 12

13 z z ω x y ω x y ω x x x Figure 5: Definition of a right handed rotation about the x axis x y x = Then note that R x R z R z R x. cos(ω ) sin(ω ) 0 sin(ω ) cos(ω ) The rotation groups in 2 (SO(2)) and 3 SO(3)) dimensions are isomorphic, having the same commutation relations and structure constants. The rotation about the ẑ axis in 3-D is written; R z = cos(ω) sin(ω) 0 sin(ω) cos(ω) This contains the subgroup SO(2), as is obvious from the earlier discussion concerning irreducible subgroups obtained by diagonalizing the matrix representation. Continuous group algebra is equivalent to Lie algebra, which involves commutators as will be seen below. Thus as an example, consider the rotation group SO(2). As above a group element can be written; ( ) cos(ω) sin(ω) R(2) = = I cos(θ) + iσ sin(ω) cos(ω) z sin(θ) = e iσzθ In the above, I is the 2-D identity matrix, and σ z is the Pauli spin matrix. Note that R(θ 1 )R(θ 2 ) = R(θ 1 + θ 2 ) as long as the rotations are about the same axis. Then; x y z R(s j ) = e iǫ js j 1 + iǫ j s j (1/2)(ǫ j s j ) 2 + Here s i are the group generators. Then R 1 (s i ) = R( s i ) since RR 1 = 1 R 1 (s j ) = 1 iǫ j s j (1/2)(ǫ j s j )

14 However, suppose an infinitesmal transformation ω 0 so that sin(ω) ω, cos(ω) 1, and ωω 0. The rotation matrix is then; R x = 0 1 ω 0 ω 1 We find that in this case; 1 ω 0 R x R z = ω 1 ω = R z R x 0 ω 1 The rotation operator is unitary. Therefore, when operating on a wave function U = e i ω L = I + i ω L ( ω L)( ω L)/2 In the above I is the unit matrix, and the meaning of the exponential operator is shown by its series expansion. Remember one must maintain the order of the operators. To obtain the infinitesmal operator matricies written above, take the 1 st two terms in the expansion assuming ω is small and use the L operators defined below. L x = L y = L z = Now we note that L x L y L y L x = L z and in general one obtains the commutation rules [L i, L j ] = L k These form the generators of a continuous group, SO(3), and define the Lie algebra. We have already noted that this group is homomorphic to the group SU(2). The above operations are not the lowest representation (irreducible representation) of this group. The operators of the irreducible representation are the Pauli spin matricies. When one adds angular momentum operators, for example, one obtains a higher order representations of the SU(2) group. Return to the unitary operation; U = e iω il i, where the L i are the generators as described above. This transformation is to be interpreted as; 14

15 U = I + iω i L i ω 2 i L 2 i/2 + Up through 2 nd order in the expansion; I + ω 2 / = ω 2 / ω 2 /2 Although this only shows the series to 2 nd order, one can at least see the beginning convergence of the series to the harmonic forms sin(ω) and cos(ω). Thus an infinite number of infinitesmal rotations reproduce a finite angle rotation. 8 Structure constants of the Lie group Look at a series of small rotations which can be written, R 1 i R 1 j R i R j. As infinitesmal operations this is; R 1 i R 1 j R i R j (1 iǫ i s i )(1 iǫ j )(1 + ǫ i )(1 + ǫ j ) (1 + ǫ i ǫ j [s j, s i ]) = 1 + c k ij s k k This operation remains near but not equal to the identity because the operators do not commute. The operation does result in an element in the group, however. ([s j, s i ]) = k c k ij s k The c k ij are the structure constants of the group and form the Lie algebra and s k the generators of the group. Lie groups are formed in higher dimensions, for example SU(3). There are more group generators in higher dimensions, in the case of SU(3) there are 8 M k, but the commutator relations remain. Suppose generators of the rotation group SO(3) are M i. The commutator relations are; [M i, M j ] = ic ijk M k As above, the structure constants are c ijk of the Lie group algebra. An irreducible representation in SU(2) of the generators of this group are the Pauli spin matricies. Every 2-D matrix with zero trace and be represented by a linear combination of Pauli sipn matricies. Then using the Jacobi identity for the commutators; [[s i, s j ], s k ] + [[s i, s k ], s i ] + [[s k, s i ], s j ] = 0 15

16 The structure constants are constrained to be; [c m ijc n mk + cm jk cn ml + cm kl cn mj] = 0 m For example, look at a representation for SU(2). ( ) ( ) a b a U = b a U 1 b = b a U 1 U = U = a a + b b = 1 In SO(3) this forms the 3 Cayley-Klein parameters which are connected to rotations about the 3-D set of axes in coordinate space. Observe that the transformation U transforms a 2-D spinor as; ( ) ( ) ( ) α a b α = b a β β α = aα + bβ β = b α + a β Thus SU(2) is homomorphic to the rotation group in 3-D. A one-to-one correspondance between the elements is an isomorphism. If the elements obey the same multiplication table the groups are homomorphic. Now look for irreducible representations of SU(2). This requires extensive algebra. Begin by describing the above transformation on the spinor in polynomial form. Without following the development one finds a unitary representation for the rotation group and identifies; a = e iα/2 cos(β/2)e iγ/2 b = e iα/2 sin(β/2)e iγ/2 In matrix form this is; ( ) ( ) ( e iα/2 0 cos(β/2) sin(β/2) e iγ/2 0 0 e iα/2 sin(β/2) cos(β/2) 0 e ( ) iγ/2 e iα/2 cos(β/2)e iγ/2 e iα/2 sin(β/2)e iγ/2 ) = e iα/2 sin(β/2)e iγ/2 0 e iα/2 cos(β/2)e iγ/2 e iα/2 The first matrix on the left is a rotation about the ẑ axis, the second a rotation about the ŷ axis and the third a rotation about the ˆx axis. The correspondance of the elements representing the rotation about the ẑ axis is; 16

17 U z = ( e iα/2 0 0 e iα/2 ) cos(α sin(α) 0 sin(α) cos(α) However, the correspondance is not one-to-one. As α goes from 0 to 2π, α/2 goes from 0 to π. Thus SU(2) and SO(3) are homomorphic but not isomorphic. Every 2-D unitary matrix of determinant 1 corresponds to a 3-D rotation. The a, b amy be muliplied by a phase to obtain the standard representation of the rotation elements; D j (α, β, γ) m,m = λ ( 1) λ (j + m)!(j m)!(j + m )!(j m )! (j m λ)!(j + m λ)!λ!(λ + m + m)! e im α cos 2j+m m 2λ (β/2) sin 2λ+m m (β/2) e mγ This representation is (2j + 1) dimensional. It may be shown that; D l 00 = P l(cos(β)) Dm0 ( l = 4π ( 1)m 2l + 1 )Y l m (β, α) D0m l = ( 4π 2l + 1 )Y l m (β, α) The components transform among themselves as; Yl m (θ, φ ) = m Yl m (θ, φ) Dmm l (R) This makes connections to integrals involving the rotation operators and their addition (Clebsh-Gordon coefficients). 9 Classical interpretation of Clebsch-Gordon coefficients Recall that in many cases of a composite system, one must add angular momentum components to obtain the system angular momentum. This for 2 angular momentum terms one has; Yl m (θ, φ)yl m (θ, φ) = (2l + 1)(2l + 1)(2L + 1) 4π ( ) ( ) LM l l L l l L m m YL M M (θ, φ) The terms; 17

18 z L m+1 m m 1 l P l Length to be projected on z Figure 6: The classical geometry for vector addition of angular momentum. The Clebsch- Gordon coefficient is the probability of finding the projection of the addition of the vectors on the z axis equal to M ( l l L ) m m M are the Wigner 3-J symbols and are related to the Clebsch-Gordon coefficients by; ( ) l l L m m = ( 1) M l l M 1 (l, m, l, m L, M) 2L + 1 Classically the Clebsch-Gordon coefficients are the probability that the z components of l, and l are m and m when the angular momentum vectors add to L and the z component is m + m = M. This is illustrated in Figure 6. The probability is found to be; ( ) l l L P = (2L + 1) [ ] Projecting angular momentum symmetry when combining states As an example of combining angular momentum consider 2 particles in a central potential well. This is the way initial wave functions are obtained for multi-particle states in quantum mechanics. These wave functions are then used in a diagonalization process to include the residual inter-particle interaction. To simplify, assume 2 particles in a cnetral well. Neglect the residual potential, we only need here the wave fumctions before diagonalization. There are 2 wave functions, ψ i ( r i ). In these wave functions the angular momentum eigenfunction is Yl m and the spin eigenfunction is j µ. The magnitude of the nuclear (or electron spin 18

19 depending on the problem of interest) is J = 1/2. ψ i = A i R l, i (r i ) Y m i l i j µ i i The product wave function is Ψ = A R l1 (r 1 ) R l2 (r 2 ) Y m 1 l 1 Y m 2 l 2 j µ 1 1 jµ 2 2 Then combine the angular momentum terms. Y m i l i j µ i i = (l i, m i, j i, µ i J i, M i ) J M i i Rewrite the product wave function in an expansion of angular momentum terms as; Ψ = A R l1 (r 1 R l2 (r 2 ) ( 1) l 1 j 1 +M 1 +l 2 j 2 +M 2 ( ) ( ) (2J1 + 1)(2J 2 + 1) l1 j 1 J 1 l2 j 2 J 2 J M 1 1 J M 2 2 m 1 µ 1 M 1 m 2 µ 2 M 2 Ψ = A R l1 (r 1 ) R l2 (r 2 ) ( 1) l 1 j 1 +M 1 +l 2 j 2 +M 2 +J 1 J 2 +M ( ) ( ) ( ) (2J 1 + 1)(2J 2 + 1)(2J + 1) l1 j 1 J 1 l2 j 2 J 2 J M 1 1 J M 2 J1 J 2 J 2 m 1 µ 1 M 1 m 2 µ 2 M 2 M 1 M 2 M The 3-j symbols can be contracted into 6-j symbols. As illustrated in the handout. 11 Representations of continuous groups We developed the definition of a continuous group in terms of a unitary rotation in 3-D coordinate space. The infinitesmal operators of the rotational transformation were found to have an algebra defined by the commutation rule [S i, S j ] = i ǫ ijk S k. Because the group is continuous, an infinite number of such infinitesmal transformations must be applied to reproduce a finite transformation. The lowest representation of this algebra, SU(2), is obtained by the 2-D Pauli matricies. There are 2 eigenvalues ±1, and these matricies are unitary and traceless. Now there are N-dimensional representations of this group, i.e. we can find N dimensional matricies that satisfy the algebra, where N is an arbitrary integer > 0. If N = 3 we found that the regular representation of this group is a set of 3 3 matricies. Different representations can be specified by the eigenvalue, S(S + 1), of S 2. here S is the maximum eigenvalue of S 3. In the case of the irreducible representation S 3 = 1/2 so that the eigenvalue of S 2 is 3/4. The value of S 3 for the regular representation is 1 so the eigenvalue of S 2 is 3. Therefore, rotational symmetry for various angular momentum values L is described by a representation of SU(2). 19

20 Remember that rotational symmetry requires that the system is independent of the spin projection on the z axis, m value. However, the energy will depend on the total value of the angular momentum. Thus, while the interaction is independent of the projection, S 3, it can depend on S 2. Indeed, the higher the eigenvalue of S 2 the higher the self energy (mass) of the particle. 12 Isospin Symmetries other than rotations can behave in a way similar to rotations in coordinate space. One of these symmetries is Isospin, which is an unfortunate misnomer. The only correspondance of Isospin to angular momentum is that it has the same algebraic structure, SU(2). The underlying reason that a particle system is invarient under a rotation in Isospin space is that the u and d quarks have almost the same mass and interact similarly. We will discuss this in more detail later. For the moment, we observe an experimental fact that the strong interaction between neutrons and protons is almost identical. That is, without an electromagnetic force, protons and neutrons would be indistinguishable. The small mass difference between a neutron and a proton would be assigned to the self energy of the electromagnetic field. This begs the question about summetry and symmetry breaking - something to be discussed later. Thus we have 2 particle states for this species of hadrons which are indistinguishable, just as 2 spin states are indistinguishable for rotational invariance. The algebra is the same as rotations, and expressed by the group, SU(2). The lowest representation has 2 states, but we find a hiearchy of representations. While the interaction is independent of the projection, T z it can depend on T 2 just as for the case of angular momentum. Here we define the Isospin operator T corresponding to S and T 3 coresponding to S 3. We write a nuclear wave function as; ψ N = C(l, m)r l,τ Y m l (θ, φ) σ 1/2 τ 1/2 Here τ is the spin vector similar to the angular momentum spin vector σ. For a multinucleon wave function, the angular momentum is combined to provide an overall symmetry by using the Clebesh-Gordon coefficients which force the wave function to be a representation of SU(2). In a similar way the isospin vectors are combined using Clebesh-Gordan coefficients. As an example consider the wave function for the deuteron. It is composed of a neutron and proton with spin of 1 and 0 angular momentum. Combining two spin 1/2 systems there are two possibilities, spin 1 or spin 0. Spin 1 gives a symmetric wave function upon interchange of the nucleons and spin 0 gives an anti-symmetric wave function. In a similar way we combine the isospin vectors to produce an isospin of 1 or 0. To obtain an overall antisymmetric wave function as is needed for a 2-baryon system a spin 0 must be coupled with an isospin 1 wave function and a spin 1 must be coupled with an isospin 0 wave function. Because the deuteron spin is known to be 1 the later wave function represents the deuteron ground state, composed of a neutron and proton (isospin 0). There are no bound 20

21 states for the isospin 1 system consisting of a spin 0 system composed of pp, np, nn (i.e. an isospin 1 system) 13 Gellmann-Nishijima relation Gell-mann and Nishijima found a connection between Baryon number and charge, T 3. Q = [(1/2)B + T 3 ] In this equation B is the baryon number, Q the charge and T 3 the 3 rd component of isospin. As an example consider the ++ (1232), a resonant state as defined above. This state has a charge Q = 2, baryon number = 1, and T 3 = 3/2. This equation is valid for nucleon states but not for hyperons. Note the commutation relations [Q, T 3 ] = 0 and [Q, T] 0 showing that isospin is not invarient when applying the electromagnetic interaction. 14 Combining SU(N)symmetries Suppose we have a system that is symmetric under 2 or more SU(N) symmetries. These are combined, as previously indicated, so that the overall system can be represented by a SU(N) representation. The dimensions of this representation can be obtained by a graphical technique called Young s tableaus. Suppose we have a system that has a symmetry SU(N) SU(N ). Let a box symbol denote the fundamental representation of SU(N). A column of (N - 1) boxes is the conjugate representation. One builds a tableau by combining the boxes using the rules; 1. Form the tableaus of the two fundamental representations 2. Choose one of the representations as the base 3. Label rows of boxes as shown in the figure below 4. Construct correct tableaus (length of a row length of the row above) 5. Keep adding boxes on this row until exhausted, then move to the row below 6. Two boxes with same label cannot be in the same column 7. Keep only N + N rows 8. Keep only tableaus with row 1 > row2 > row 3 21

22 N N+1 N+2 N+3 N+4 Hook N 1 N N+1 N+2 N+3 N 2 N 1 N 3 Figure 7: An example of a Young s Tableau SU(2) X SU(2) X SU(2) 2 X 2 X 2 X Hook X Figure 8: An example of a Young s Tableau Fill the boxes with the numbers shown in Figure 7. The dimension of a tableau is obtained by evaluating the product of the boxes divided by the number of hooks. Hooks are counted as shown in the figure. The dimension of a tableau is N! where the number of hooks is determined by drawing no.hooks all lines through a row of boxes and then down through a column and counting the number of boxes that are crossed and multipling these numbers together. Consider the example of finding the symmetry and dimension of SU(2) SU(2) SU(2) These are combined in the figure in steps. The lowest tableau is not possible so it is discarded. There are three possible tableaus having dimensions; 22

23 (2)(3)(4) (1)(2)(3) = 4 (2)(3) (1)(3)(1) = 2 15 Example Suppose the example of the deutron which we solve in relative coordinates. This example has 2 interacting nucleons each with spin, j i, and isospin, τ i. To a good approximation, the deuteron has no amgular momentum term, Y0 0. Thus combine the spin and isospin to obtain; j µ 1 1 j µ 2 2 = (j 1, µ 1, j 2, µ 2 J, M) J M τ ν 1 1 τν 2 2 = (τ 1, ν 1, τ 2, ν 2 τ, ν) τ ν Ψ = AR(r) (j 1, µ 1, j 2, µ 2 J, M) (τ 1, ν 1, τ 2, ν 2 τ, ν) J M τ ν The spin and isospin can also be combined but we do that in a different way. The spin and isospin wave functions each have SU(2) symmetry. Thus we expect the deuteron wave function to have the form; SU(2) SU(2) SU(2) SU(2) Consider the first two SU(2) terms represent the spin of the 2 nucleons, and the second their isospins. Thus we expect 6 possibilities. Two neutron-neutron states, one with spin 0 and one with spin 1 Two proton-proton states, one with spin 0 and one with spin 1 Two neutron-proton states, one with spin 0 and one with spin 1 This symmetry is combined using Young s tableaus, Figure 9. Because of the Pauli excliusion principle the nn and pp state with spin 1 cannot exist. There are 4 states with spin 0. The interaction in this spin state is found to be weaker that that for the spin 1 case so these states are unbound. There is one remaining state with spin 1. This np state forms the deutron. It has dimension of 3 representing the 3 projections of the spin on the z axis. 23

24 SU(2) X SU(2) X SU(2) 2 X 2 X 2 X 2 2 X 2 X X 2 X 2 X X nn and pp spin 1 Dimension 5 nn spin 0 Dimension 3 pp spin 0 Dimension 3 J Τ X pn spin 0 Dimension 1 pn Spin 1 Dimension 3 pn spin 0 Dimension 1 Figure 9: Young s Tableau for two nucleons showing the dimensional representations 24

25 16 More on the Gellmann-Nishijima relation In the last lecture we found that the Gell-mann and Nishijima relation related Baryon number, charge, and the third component of isospin. Q = [(1/2)B + T 3 ] But this relation was not valid for hyperons. Now we also know that hyperons are different from nucleons and nuclear resonances. They have long lifetimes, which are more like lifetimes associated with the weak interaction. Thus Pais suggested that these paticles possessed another quantum number which was called strangeness, S. The Gell-mann and Nishijima relation was then extended to include strangeness through the expression; Q = [(1/2)(B + S) + T 3 ] An assignment of strangeness or anti-strangeness was assigned to each of the hyperons and hyperon resonances. The quanty B + S is called hypercharge, Y, something we used earlier in the lecture. 17 Extension of SU(2) to SU(3) The fact that neutrons and protons can t be distinguished under the strong interaction is now extended to include hyperons. The fundamental doublet; ( ) u d This is extended to include the lowest mass hyperon Λ, to form a triplet, forming the lowest fundamental representation of the special unitary group, SU(3). u d Λ This symmetry is violated more strongly than that of isospin. One certainly observes that the mass of the Λ is much larger than that of the nucleon, 1115 to 939 MeV. Still this triplet of states is an approximate symmetry. In this case we have an operator, U, that transforms a triplet state, ψ into an equivalent state; ψ = Uψ The operators U must be 3 3 unitary, traceless matricies as are the Pauli matricies in 2 dimensions. There are 8 independent Hermitian, traceless 3 3 matricies λ i, and they satisfy 25

26 0 K +1 Y K + π π π η +1 I z 0 K 1 K _ SU(3) Octet for J P= 0 Mesons Figure 10: An SU f (3) weight diagram for mesons the commutation rules; [λ i, λ j ] = i ǫ ijk f ifk λ k where the f ifk are the structure constants of the SU(3) group. A graphical representation of the group structure is a weight diagram of the group elements. This is illustrated by the weight diagram for mesons and baryons in Figures 10 and 11, respectively. It is important to note here that the application of the SU(3) group applies to flavor. This is a reference to a class of quark families which will be discussed later. The more commonly known application of SU(3) applies to color which is a different quantum number of quarks that will also be discussed later. The Strong interaction, which includes nucleons and hyperons, is approximately symmetric under SU f (3) 18 Parity The normal modes of a string have either even or odd symmetry. This also occurs for stationary states in Quantum Mechanics. The transformation is called partiy. We previously found for the harmonic oscillator that there were 2 distinct types of wave function solutions characterized by the selection of the starting integer in their series representation. This selection produced a series in odd or even powers of the coordiante so that the wave function was either odd or even upon reflections about the origin, x = 0. Since the potential energy function depends on the square of the position, x 2, the energy eignevalue was always positive and independent of whether the eigenfunctions were odd or even under reflection. In 1-D parity is the symmetry operation, x x. In 3-D the strong 26

27 Ξ dss 0 1 Y Ξ + uss Σ 1 0 dds Λ Σ uds + Σ uus +1 I z udd N +1 P uud Baryon Octet Figure 11: An SU f (3) weight diagram for baryons Even Odd interaction is invarient under the symmetry of parity. Parity is a ciscrete symmetry, U(1), which is homomorphic to charge symmetry. r r Parity is a mirror reflection plus a rotation of 180, and transforms a right-handed coordinate system into a left-handed one. Our Macroscopic world is clearly handed, but handedness in fundamental interactions is more involved. Vectors (tensors of rank 1), as illustrated in the definition above, change sign under Parity. Scalars (tensors of rank 0) do not. One can then construct, using tensor algebra, new tensors which reduce the tensor rank and/or change the symmetry of the tensor. Thus a dual of a symmetric tensor of rank 2 is a pseudovector (cross product of two vectors), and a scalar product of a pseudovector and a vector creates a pseudoscalar. We will construct bilinear forms below which have these rotational and reflection characteristics. 19 Time reversal Time reversal is the mathematical operation; 27

28 t t; with exchange of initial and final states. Macroscopically T is not a good symmetry. However, For Quantum Mechanics; T HT = H; Hψ = i t ψ; THψ = [Tψ]; H[Tψ] = i [Tψ]. Thus Ψ and [T ψ] are not equivalent, and T requires t t and i i. However, one constructs observables in Quantum Mechanics by bilinear forms, ( i.e. by products two operators and wave functions, as discussed later) so that microscopic time reversibility holds. 20 Charge conjugation Charge Conjugation changes a particle to its anti-particle, but without change to its dynamical variables. The symmetry is based on the asumption that for every particle there is an antiparticle which has Q Q, B B, L L, etc. An eigenstate of C must have: Q = B = L = S 0. Thus a π 0 is an eigenstate of C but K 0 is not since it contains S, a quark of the 2 nd generation. The strong and electromagnetic interactions are invarient under C. Under the weak interaction the operation C is not a good symmetry. 21 The operations of P and T The operations of reflection and time reversal in classical systems is shown in Table below. 28

29 µ + e + _ C Violation in ν e νµ e + e _ ν e ν µ s s µ + s µ s µ s C s CP s s _ s s s s ν e ν µ e _ ν νµ e Name P T Time + - Position - + Energy + + Momentum - - Spin + - Helicity - + Electric Field - + Magnetic Field + - Obviously some parameters are invarient but some change sign under this combined operation. 22 The operations of P and C The operation of CP is composed of the simultaneous operations of C and P. Suppose one wishes to distinguish a galaxy from an anti-galaxy. It is not sufficient to find C violation but one needs CP violation as well. The weak interaction violates C and P but CP is experimentally conserved except for flavor changing decays. In flavor changing weak decay CP is not preserved. The K 0 and K 0 are eigenstates of strangeness but not of CP. However states of the weak interaction ( as presently defined) are invarient under CP. Thus the two possible CP eigenstates of the K 0 (K 0 ) have different masses and decay widths. However it was found that the decay of CP eigenstates does not preserve CP. 29

30 C and P Transformations for π + µ + ν µ s ν π + µ µ + µ + π + s P ν µ C C CP _ ν µ π π _ µ P µ ν µ s 23 The operations of P and T The operations of reflection and time reversal in classical systems is shown in Table below. Name P T Time + - Position - + Energy + + Momentum - - Spin + - Helicity - + Electric Field - + Magnetic Field + - Obviously some parameters are invarient but some change sign under this combined operation. s 24 Bilinear forms of Dirac wave functions We recall that a Dirac wave function has 4-components, and that γ, α and β are 4 4 matricies used in the dirac equation. As an aside note that the current j = cψ αψ leads to an expectation value of the velocity. We write the following bilinear forms that have the various listed transformation properites; In the above, ( 0 σ γ = σ 0 ) 30

31 Bilinear Form Transformation Property ψ ψ ψ γ n ψ ψ γ 5 ψ ψ γ 5 γ n ψ Scalar Vector Pseudoscalar Pseudovector State Energy Helicity Chirality 1 > > < < ( ) I 0 γ 0 = 0 I ( ) 0 I γ 5 = i I 0 where σ are the Pauli spin matricies, I is the 2 2 idenity matrix, and γ 5 = γ 1 γ 2 γ 3 γ 5. Note that the γ i are the components of a relativistic 4-vector, and ψ is the adjoint of the Dirac wave function ψ. The helicity of the wave function is defined as the direction of the particle spin vector relative to the momentum vector. It measures the handedness of a particle and is a pseudoscalar invarient under T. Σ = σ p/ p The chirality operator, γ 5, operates on the helicity states to produce the chirality of the state. We then find the helicity and chirality of the eigenvectors for the various Dirac states For E > 0 states, a spatial reflection inverts the momentum vector and changes the sign of the helicity. The Chirality of a state is optained by the projection operator; 31

32 P ± = 1/2(1 ± iγ 5 ) The projection operator has the properties; P + + P = 0 P 2 ± = 1 P + P = P P + = 0 25 The operations of P, C, and T Conservation of the simultaneous application of C, P, and T is expected under very general conditions. In all Lorentz invarient quantum field theories, CPT is a good symmetry. This means that if CP is violated then T must be violated as well. Direct searches for T violation are difficult as null experiments are not easily designed. 26 Zero mass equation In the case of zero mass the Dirac equation has the form; [i t + iγ 0 γ ]ψ = 0 Which looks like ; ( 0 σ [E σ 0 ) ]ψ = 0 If we divide the 4-component Dirac wave function into 2 two component wave functions described by upper and lower components ψ u and ψ l respectively, the zero mass dirac equation forms 2 equations; Eψ u σ p ψ l = 0 Eψ l σ p ψ u = 0 Then a specific chirality state is not a state of specific parity, and can be described by a 2-component wave function; 32

33 Symmetry Operations + P T + C + 33

34 and ; ( ) 0 ψ = P ψ = φ ( ) φ ψ + = P + ψ = 0 γ 5 ψ = ψ γ 5 ψ + = ψ + Thus chirality is a good symmetry for massless particles. It represents the direction of the spin relative to the momentum vector, and divides masselss Dirac states into left and right handed doublets. 27 Lagrangian The lagrangian is a Lorentz scalar, and is composed of bilinear forms in a way to make the scalar. Thus for example a vector form must be contracted (dot product) with a vector. Two pseudo-scalars can be multipled together, etc. 28 Quarks Historically, spectroscopy of the baryons and mesons led to a classification scheme of these particles in terms of their spin, parities, and a quantum number called flavor. We have already discussed nucleon and strangeness quantum numbers which we now identify by different flavors. In order to match the spectroscopy of the states we propose to construct a baryon of 3 substructures which are called quarks. Obviously quarks will have some multiple of 1/3 of the electronic charge and have baryon number of 1/3. The introduction of 3 possible states (proton like, neutron like, and strangeness like) gives an SU(3) symmetry which for 3 substructures has the symmetry SU(3) f (flavor in tis case); SU(3) SU(3) SU(3) Apply Young s Tableau to optain the dimensions 10 s + 8 m,s + 8 m,a + 1 The sub-labels represent the symmetry of the states. Note that the final state must be anti-symmetric upon exchange of quarks because they are Fermions. The spin wave function could presumably provide the remaining component to obtain the overall anti-symmetry. The symmetric 10 dimensional representation presents a problem. The weight diagram is shown in Figure 12 34

35 p + J = 3/ * Σ * + Σ * 0 Σ Ξ * Ξ * 0 Ω Figure 12: The ten dimensional representation of SU(3) One of the components of this representation is the ++. The substructure wave function is symmetric and the spin wave function must also be symmetric because the spin of the is 3/2. The only way to preserve ani-symmetry is to introduce another quantum number, color, which can be anti-symmetrize the wave function. Thus initially we used SU f (3). Now the SU(3) symmetry is the color symmetry of the quarks. There are 3 colors which are composed to provide and anti-symmetrize a color component of the wave function having dimension Classical Electromagnetism Maxwell s equations define electromagnetism; E = ρ/ǫ B = 0 E = B t B = µ J + (1/c 2 ) E t ρ Charge is conserved as is expressed by the equation of continuity. t + J = 0 with J = ρv. This can be written in a Lorentz invarient form by reconizing that a 4-vector can be formed. Thus; x µ = ( x, t); and j µ = ( j, ρ) 35

36 This allows a 4-divergence of the form x µ = (, t ) The electromagnetic fields, E, B, can be expressed in terms of potentials. Thus B = A and E = φ A t A = (µ/4π) d 3 x ( J r ) E = (1/4π) d 3 x ( ρ r ) The potentials form a Lorentz 4-vector ( A, φ). Not only are Maxwell s equations determined by the 4-vector, but we take these potential forms as more fundamental than the fields themselves, since potentials are used in the Lagrangian formulation. (also see the Bohm-Aharonov effect described below) The fields are components of a Lorentz 4-component tensor of rank 2. F µ ν = µ A ν µ A ν = 0 E x E y E z E x 0 B z B y E y B z 0 B x E z B y B x 0 Thus in fully covarient form maxwell s equations are written, µ F µν = j ν for the inhomogeneous eqations λ F µν + ν F λµ + µ F νλ 0 for the inhomogeneous eqns. However, the 4-vector potential is not unique as the same field tensor is obtained under the potential transformation; A µ A µ + µ χ = ( A χ, φ + χ t ) Here χ is an arbitrary scalar function of position and time. This is called a gauge transformation. Now for the transformation to keep the 4-vector potential relativisticly invarient, the transformation must obey a restricted class of gauge transformations (Lorentz gauge) such that; µ A µ = 0 2 χ (1/c 2 ) 2 χ t 2 = 0 36

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